with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.

The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact,

, and

When the alternating factor (–1)n is removed from these series one obtains the hyperbolic functions sinh and cosh used in calculus.

For integer or positive index α the Bessel function of the first kind may be defined with the alternating series

Proof: Suppose the sequence converges to zero and is monotone decreasing. If is odd and , we obtain the estimate via the following calculation:

Since is monotonically decreasing, the terms are negative. Thus, we have the final inequality .Similarly it can be shown that . Since converges to , our partial sums form a Cauchy sequence (i.e. the series satisfies the Cauchy criterion) and therefore converge. The argument for even is similar.

Proof: Suppose is absolutely convergent. Then, is convergent and it follows that converges as well. Since , the series converges by the comparison test. Therefore, the series converges as the difference of two convergent series .

In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. One of the oldest techniques is that of Euler summation, and there are many modern techniques that can offer even more rapid convergence.